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1 [CT041]

[CT041]

[00250] Formation of delta shock waves and vacuum states in the vanishing pressure limit of the Riemann solution to the isentropic Euler system for logarithmic equation of state with the Coulomb-like friction term

Session Date & Time : 4D (Aug.24, 15:30-17:10)
Type : Contributed Talk
Abstract : We investigate the limiting behavior of the Riemann solution to the isentropic Euler equations for logarithmic equation of state with the Coulomb-like friction term. The formation of vacuum state and delta shock waves are identified and analyzed when the pressure vanishes. Unlike the homogeneous case, the Riemann solution is no longer self-similar. We prove that the Riemann solution of the isentropic Euler equations for logarithmic equation of state with friction term converges to the Riemann solution of the zero-pressure gas dynamics system with a body force when the pressure vanishes.
Classification : 35L65, 35L67, 35L45
Author(s) :
- Anupam Sen (Post Doctoral Fellow at Centre for Applicable Mathematics, Tata Institute of Fundamental Research)

[00434] Three pieces Riemann problem for $2$-D full Euler system in the Noble-Abel gas

Session Date & Time : 4D (Aug.24, 15:30-17:10)
Type : Contributed Talk
Abstract : We present Riemann problem governed by $2$-D full Euler system in the Noble-Abel gas. Riemann data, consisting three constants, are distributed in three distinct regions with an assumption that two adjoining regions can be connected by only one planar elementary wave. We present criteria for existence of different configurations of elementary waves for isentropic, as well as full, Euler system. We also discuss the effect of the Noble-Abel gas and the angle of regions on elementary waves and corresponding stream curves. Note: The present article has been published on 17 May 2022 in the journal ''Mathematical Methods in the Applied Sciences''/ Volume 45, Issue 16 with DOI:10.1002/mma.8377.
Classification : 35L65, 35L67, 35Q30, 35Q31, 35Q35, Hyperbolic Conservation Laws, Shocks and Singularities for Hyperbolic equations, Navier-Stokes equations, Euler equations, PDEs in connection with fluid mechanics.
Author(s) :
- Harsita Srivastava (Dr. B. R. Ambedkar National Institute of Technology Jalandhar, Punjab)
- M. Zafar (Dr. B. R. Ambedkar National Institute of Technology Jalandhar, Punjab)

[00482] Delta Shocks and vacuum states in the Riemann solutions of Chaplygin Euler equations as pressure and magnetic field drop to zero

Session Date & Time : 4D (Aug.24, 15:30-17:10)
Type : Contributed Talk
Abstract : The aim of the present study is to solve the Riemann problem of isentropic magnetogasdynamics equations for a more realistic version of the extended Chaplygin gas model. The analysis demonstrates that under some special circumstances, delta shock and vacuum appear in the solution, describing the phenomena of concentration and cavitation, respectively. By examining the limiting behavior, it is obtained that solutions coincide with corresponding Riemann solutions of the transport equations when both the magnetic field and pressure drop to zero. PS: This paper has been published in J. Math. Phys. 63, 121505 (2022); https://doi.org/10.1063/5.0132580
Classification : 35L65, 35L67, 35L60
Author(s) :
- Priyanka . (Dr. B.R. Ambedkar NIT Jalandhar)
- M. Zafar (Dr. B. R. Ambekdar NIT Jalandhar, Jalandhar, India)

[00750] Bayesian inverse problems for some hyperbolic conservation laws

Session Date & Time : 4D (Aug.24, 15:30-17:10)
Type : Contributed Talk
Abstract : We study some inverse problems for hyperbolic conservation laws. Given observations of the entropy solution, we consider the problem of identifying the initial field or the flux function. Due to shockwaves, direct observations of the entropy solution are not "regulated" enough to fit in the Bayesian framework in Stuart (2010). To get round this, we propose a new approach by studying the trajectories for hyperbolic conservation laws and exploring their existence, uniqueness and stability.
Classification : 35L65, 35R30, 62F15, Partial Differential Equations, Inverse Problems
Author(s) :
- Duc-Lam Duong (LUT University)
- Duc-Lam Duong (LUT University)
- Masoumeh Dashti (University of Sussex)

[01144] Non-Local Conservation Laws with Discontinuous velocity Vector Field

Session Date & Time : 4D (Aug.24, 15:30-17:10)
Type : Contributed Talk
Abstract : Stability of entropy solutions of nonlocal scalar conservation laws: \begin{eqnarray*} u_t + (s(x)u \nu(u*\xi))_x&=& 0\,\quad\quad \quad \quad \quad \text{for}\,\,\,(t,x) \in (0,\infty)\times \mathbb{R}, \\ u(0,x)&=&u_o(x) \quad\quad \quad \,\, \text{for}\,\,\,x \in \mathbb{R}. \end{eqnarray*} where, $\nu,\xi \in (C^2 \cap W^{2,\infty})(\mathbb{R}), s$ is a positive step function and \begin{eqnarray*} (u*\xi)(t,x) &:=&\int_{x}^{x+\eta}u(t,y)\xi(y-x) d y, \end{eqnarray*} will be presented. Estimates of the dependence of solution with respect to $\xi$, $s$,$\nu$ and $u_o$ are established from the entropy condition are established.
Classification : 35L65, 35B44, 65M06, 65M08, 35A01
Author(s) :
- Aekta Aggarwal (IIM Indore)

[01148] Error Estimates for Conservation Laws with Discontinuous Flux

Session Date & Time : 4E (Aug.24, 17:40-19:20)
Type : Contributed Talk
Abstract : We consider a certain class of conservation laws with discontinuous flux, where, the set of spatial discontinuities of the flux can be possibly infinite with accumulation points. It is shown that the monotone finite volume schemes converge to the entropy solution, at the rate of $\sqrt{\Delta x}$ in $L^1$ norm. Numerical experiments are presented to illustrate the theory.
Classification : 35L65, 35B44, 35A01, 65M06, 65M08
Author(s) :
- Ganesh Vaidya (TIFR CAM, Bengaluru)

[02614] Convergence of a Second-Order Scheme for Nonlocal Traffic Flow Problems

Session Date & Time : 4E (Aug.24, 17:40-19:20)
Type : Contributed Talk
Abstract : In this work, we focus on the construction and convergence analysis of a second-order numerical scheme for traffic flow models that incorporate non-local conservation laws to capture the interaction between drivers and the surrounding density of vehicles. Specifically, we combine MUSCL-type spatial reconstruction with strong stability preserving Runge-Kutta time-stepping to devise a fully discrete second-order scheme for these equations. We show that this scheme satisfies a maximum principle and obtain bounded variation estimates. Also, the scheme is shown to admit L1- Lipschitz continuity in time. Subsequently, employing the Kolmogorov's theorem with a modification and using a Lax-Wendroff type argument, the convergence of this scheme to the entropy solution of the underlying problem is established. Numerical examples are presented to validate our theoretical analysis. Additionally, we extend our analysis to two dimensional non-local problems, for which we present a positivity preserving second-order scheme. While first-order methods are typically reliable in computational fluid dynamics, higher-order methods can provide more accurate solutions at the same computational cost, especially for problems in two or three dimensions. Our proposed scheme thus has important implications for accurately approximating traffic flow equations, and our theoretical analysis provides a solid foundation for its practical implementation.
Classification : 35L65, 65M12, 65M08
Author(s) :
- Nikhil Manoj (Indian Institute of Science Education and Research, Thiruvananthapuram)
- Sudarshan Kumar K (IISER Thiruvananthapuram)
- GD Veerappa Gowda (Center for Applicable Mathematics, TIFR Bangalore)

[00028] Riemann problem for the Chaplygin gas equations for several classes of non-constant initial data

Session Date & Time : 4E (Aug.24, 17:40-19:20)
Type : Contributed Talk
Abstract : Using the differential constraint method, a class of exact solutions is obtained for the homogeneous quasilinear hyperbolic system of partial differential equations describing Chaplygin gas equation; these solutions exhibit linearly degenerate that leads to the formation of contact discontinuities. In fact, in this paper, we solved the gen- eralized Riemann problem through a characteristic shock(s). For several classes of non-constant and smooth initial data, the solution to the generalized Riemann problem is presented.
Classification : 35L67
Author(s) :
- Akshay Kumar (University of Hyderabad)
- Radha R (University of Hyderabad)

[01163] An infinite class of shocks for compressible Euler

Session Date & Time : 4E (Aug.24, 17:40-19:20)
Type : Contributed Talk
Abstract : We consider the two dimensional compressible Euler equations with azimuthal symmetry and construct an infinite class of shocks by establishing shock formation for a new Hölder family of so-called pre-shocks for all nonnegative integers. Moreover, a precise description of the dominant Riemann variable in the Hölder space is given in the form of a fractional series expansion.
Classification : 35L67, 35Q31, 76N15, 76L05
Author(s) :
- Calum Rickard (University of California, Davis)
- Sameer Iyer (University of California, Davis)
- Steve Shkoller (University of California, Davis)
- Vlad Vicol (New York University)

[00775] Weak Shock diffraction in a van der Waals gas

Session Date & Time : 4E (Aug.24, 17:40-19:20)
Type : Contributed Talk
Abstract : The objective of this study is to analyze the diffraction of a weak shock hitting a thin semi-infinite wedge screen. The Riemann problem is formulated for two-dimensional compressible Euler system for real gas considering the van der Waals equation of state. To study the incident and diffracted shock the self-similar flow has been considered. Also, in some regions of semi-infinite wedge screen rarefaction waves are found and studied. The real gas effects have been studied.
Classification : 35Lxx, Hyperbolic equations and hyperbolic systems
Author(s) :
- Gaurav Gaurav (IIT(BHU), Varanasi, India)